# Tagged Questions

Questions about the properties of functions of the form $\sum_{n=0}^{\infty}a_n x^n$, where the $a_n$ are real or complex numbers, and $x$ is real or complex (or more generally an element of a Banach algebra).

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### help in real analysis [on hold]

How can I use this definition to prove that $a^{\frac{1}{n}}$ converge to $1$? where a >0
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### Order of summation for shifted exponential function

I want to represent the function: $$f(x)=e^{-a(x-b)^{2}}$$ where, $0<a<1$, $x\in\mathbb{R}$, and $b\in\mathbb{R}$. As a power series for an integral I am working ...
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### Reciprocal of the sum of powers of $1/x$ [duplicate]

Incidentally, I found $$\frac{1}{\sum_{n=1} \frac{1}{x^{n}}} = (x-1)$$ where $x\ge 2$. Please direct me to how others have developed the relationship. My computer cannot compute more than X = ...
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### Find a power series representation for the function. [closed]

I'm not sure how to handle this problem. I got that the radius of convergence was 1/6, but I don't know how to represent the function as a power series. I can modify it to look like the following: x ...
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I have been self studying the classical Umbral Calculus and have been reading works and papers from Rota and Roman on the material and I have a question regarding the following. The text uses $$\... 1answer 44 views ### Tight bounds on the sum of the jth power of the first n natural number? What are the tight bounds for S_{n,j}=\sum_{k=1}^n k^j? Where O(j)=O(n^3). 2answers 54 views ### Problem calculating the sine of a matrix Given the matrix A=\begin{pmatrix}-\frac{3\pi}{4} & \frac{\pi}{2}\\\frac{\pi}{2}&0\end{pmatrix}, I want to calculate the sine \sin(A). I do so by diagonalizing A and plugging it in the ... 2answers 121 views ### Is it possible to have a power series for arctan(x) centered at 1? My Calculus 2 professor referenced that such a series is impossible, but why? I understand how to properly find the power series of arctan(x) centered at 0. 1answer 51 views ### How to determine sum of an alternating power series and to prove that sum is positive I am working on a problem involving an alternating power series as follows:$$\sum_{i=0}^{a-2} (-1)^{a+b-i-2}(a+b-i-1)x^{a+b-i-2}$$a and b is constant with 0<x<1 I would like to ... 0answers 29 views ### Quotient of Confluent Hypergeometric Functions of the 1st Kind I want to solve the following problem for x: $$\frac{\mathrm{d}}{\mathrm{d}x}\ e^{-\beta_{1}x}\,{_{1}}F_{1}[-\alpha_{1};-\alpha_{3};\beta_{3}x]=0$$ where, \alpha_{1},\... 1answer 22 views ### Radius of convergence of integral series; problem with limsup Let \sum c_k x^k be a power series with radius of convergence R. Then the integral series$$\sum_{k=0}^\infty \frac{c_k}{k + 1}x^{k+1}$$also has radius of convergence R. I'm reading Real ... 3answers 34 views ### Find the radius of convergence as well as the interval of convergence: I've done a bunch of these and was successful, but this one is proving to be troubling. I have no idea how to handle that n*sqrt(n) at the bottom. I ended up with some jank shit like this lol: Lim ... 1answer 20 views ### Find the relative width of a guitar fret There is an equation to find the position of a fret on a guitar fretboard, given the length of a string is given by \begin{eqnarray} d = s – \frac{s}{2 ^ {(n / 12)}}, \end{eqnarray} where d is the ... 2answers 27 views ### Why is \operatorname{Sech}(x) Taylor series divergent past \pi/2? Someone asked a question here about why the Taylor series of \log(1+x) diverges: Why does the taylor series of \ln (1 + x) only approximate it for -1<x \le 1? I have a similar question: why ... 1answer 65 views ### Can the series \sum_{n=0}^\infty \frac{(-1)^n}{\sqrt{2^n n!}} x^{n} be summed? [on hold] Can the following series$$\sum_{n=0}^\infty \frac{(-1)^n}{\sqrt{2^n n!}} x^{n}$$be summed? 2answers 27 views ### Find the Taylor-series expansion of a square of a rational function of a complex variable I've been trying to find the Taylor-series expansion of the following function:$$ f(z)=\left ( \frac{1+z}{1-z} \right )^2 $$az the origin : Z0 = 0. also I would like to find the region of ... 1answer 68 views ### What is the interval of convergence of : \sum_{n=1}^\infty\frac{n^n}{n!}x^n? x+ \frac{2^2x^2}{2!}+ \frac{3^3x^3}{3!}+ \frac{4^4x^4}{4!}+... Possible answers- 1.(0,1/e) 2.(1/e, \infty) 3.(2/e, 3/e) 4.(3/e, 4/e) ... 1answer 25 views ### Field and ideal notation: double bracks/parens vs single brackets/parens I'm reading some notes that has the following denotation: the set of formal power-series with coefficients in \mathbb{F}_p is denoted by \mathbb{F}_p[[t]]. the fraction field, \operatorname{... 0answers 14 views ### Sum of a converging series having Error function with a polynomial I am struggling to find the sum of the following series: k\sum\limits_{z=1}^{\infty} \frac{(z+1)^2}{4} . erfc(az) where k and a are known parameters and erfc(x) is the complementary error ... 0answers 35 views ### Deciphering the theorem of perfect powers Can someone help me understand what happened to this equation from the paper entitled Perfect Powers With All Equal digits but one...I don't understand the part when it lets a and c not equal to zero.... 5answers 1k views ### What does it mean intuitively for a Taylor Series to be centered at a specific point? I understand what a Taylor series is and how to find the Taylor series of a function. However I do not understand intuitively what it means to find a Taylor series for a specific function, centered at ... 3answers 113 views ### Series \frac{x^{3n}}{(3n)!}  find sum using differentiation Find sum of the series$$\sum_{n=1}^{\infty}\frac{x^{3n}}{\left(3n\right)!}$$using differentiation. So far I found that$$S(x)+1=S'''(x)$$but it does not help. Then I tried different interesting ... 1answer 206 views ### New series formula for \arctan(x)? I discovered this equation, but have no idea if it has been previously discovered. Please help determine if it has been previously developed. Or please prove that the equation is not correct.$$\...
In the proof that symmetric random walks end up regressing to the origin with probability $1$, I have found this didactic post on-line. In it the following two definitions are given: Probability of ...